Thanks Dear Dr.David(@dharr) you are very helpfull for me on this topic, still i am work on this topic, and i did a great achivment and keep improving the work, this mehod  need a super computer for a long range of variable of (t) i saw your last graph which is so far from exact but if you make the range of t smaller it will be so small error or maybe zero,  your code is great like always, but i don't undrestand about this part when you write (infolevel[LAD] := 2) what mean of this 2 in the first example and we can change to 3 or more? i have mine code and i have to improve it, and i am intresting in your which is so compact, i don't know this method can hundle the hard one?  and if  need help  i will ask for help

@dharr  the pdetest i did it but have some condition need analytice method which in article i have to explain all of them 

num-Ex1_pdetest-satisfy.mw

please if possible make some pretty plot for me i needed i will work on it but nothing usefull get yet
also this table can be export?

table-e2.mw

@dharr i don't know how many term should i add to get that but must be there is a code do to infinite term ...

@janhardo  try to compare with result outcome is not usefull without  showing example?

@dharr same author paper, all paper of this author are the same, and i recognize one mistake  of paper which not multiply by (I=sqrt{-1}) but step for me is more important , please chekc your gmail too

e1.mw

@dharr first don't worry about exact solution i solved a lot of this kind pde and harder than this, in thus pde have some condition about that parameter maybe becuase of that don't make it to zero, i can solve any pde and finding exact solution for you if needed, and there is a trick in physic which not all people know this,  which is if u(x,t)=f(x)*exp(#) then 

|u(x,t)|=f(x), in physic we do like that i never did that cahnge with conjugate when have power  is gonna |u(x,t)|^2=f(x)^2, i hope this is help full 
i have some 2025 article which i want to know  how you needed you want me to solve them then send to you or  you want i am uploaded the result here?

@dharr I can find a lot of exact solution is depend on analysis method in paper as you see the equation (2) and (9) are exact solution of pde, after we find the exact solution we apply laplace adomian decomposition method to get numerical solution, I saw a lot paper still fractional order remain I have to apply method for fractional order I will look for other paper which have more detail 

@dharr last step is slow and i think the author of paper becuase of that didn't mention any result just two small table , i hope we didn't done any mistake, i have to search and check for more example then i will go to harder one, thank you so much dear dr.

@acer the paper didn't  give any information about the result i will upload here in case  don't beliave me !
i know what you mean you want something like that

but they didn't give us anything becuase result are to long i think and just mention the number N which i think is iteration on number x  this is the paper   is a little bit a lot page but becuase you ask i will upload here to see 

@dharr  i have work on this amazing code and apply some other example by this becuase is so pretty, my ugly one is work thanks to you and acer, but this is amazing i will work on it more, some steps remain like table figure, erro which i will try  and search for get them, thank you so much .

@dharr  always wanna do something different, the step are amazing but i don't know why not use my ugly code in you adomian polynomial i didn't see   the same function  and devide by n! what term you have in NNv if it multiply by (a0 in you adomian must multiply by (a) too i think this is missing in Asym[n],
take a look to acer he did a great job too, just one step remain which i detect the problem between the u[2] and u[3] the adomian don't have issue at there but sign of conjugate make problem 

B-Ac.mw

@acer  i know there is some mistake by sign but problem is not the sign, you did a great job , but have a problem in my adomian polynomial  we have this term conjugate(U[i](x,t)) when you make it correct you write like U[i](x,t) with hat sign of conjugate which i try to compare i saw this and make my adomian not coming out  the problem is sign of conjugate i think which if this is by hundle all iteration it will work 

B-Ac.mw

@acer  i delete becuase in top write dublicate question and becuase of that no one answer for sure i will not lose you, i love you ... i hope to get best code

@acer  is been a week i am stuck at this step and i did for a lot of simple example without have the complex term, this is a link of question

https://www.mapleprimes.com/questions/240473-How-Substitute-Function-In-Iteration-

and again i will uploaded for you here maybe you can hundle this  you are powerfull 

i want something like this 

Ex.3.2.mw

@acer i  if you watch the first u[1] is wrong and also the answer is so different from the paper result i don't know even why he change the exponential term to function of sin and cos this is answer of paper